Hi all Need some help with integration steps. I am trying to find the integration result of d/dt P(t) +(σ+ρ)P(t) = σ This is then multiplied by the integrating factor e^((σ+ρ)t) to give the next line: e^((σ+ρ)t) d/dt P(t) + (σ+ρ) e^((σ+ρ)t) P(t)= σ e^((σ+ρ)t) We then integrate both sides with respect to t The RHS gives e^((σ+ρ)t) P(t) I dont understand how this is derived. How did we get from e^((σ+ρ)t) d/dt P(t) + (σ+ρ) e^((σ+ρ)t) P(t) to e^((σ+ρ)t) P(t). I would be grateful if I could be taken through the breakdown of the steps. Thank you kindly.
Hi Millie The integrating factor method is a reverse engineering of the product rule, so what we have in your example is: $$ u = e^{(\sigma + \rho) t} $$ $$ v = P(t) $$ Which, by applying the product rule, we can write as: $$ u dv/dt + v du/dt = d(uv)/dt $$ Then integrating both sides with respect to t yields uv. Hope this helps, Dave
